What kind of optimizer is suggested to use for binary classification of similar images?
I have consistently found Adam to work very well but to tell you the truth I have not seen all that much difference in performance based on the optimizer. Other factor seem to have much more influence on the final model performance.In particular adjusting the learning rate during training can be very effective. Also saving the weights for the lowest validation loss and loading the model with those weights to make predictions works very well. Keras provides two callbacks that help you achieve this. Documentation is at 
https://keras.io/callbacks/. The ReduceLROnPlateau callback allows you to adjust the learning rate based on monitoring a metric. Typically validation loss is monitored. If the loss fails to reduce after N consecutive epochs(parameter patience) the learning rate is adjusted by a factor(parameter factor). You can think of training as descending into a valley which gets more and more narrow as you approach the bottom. If the learning rate does not adjust to this "narrowness" there is no way you will get to the very bottom.
The other callback is ModelCheckpoint. This allows you to save the model( or just the weights) based on monitoring of a metric. Again usually validation loss is monitored and the parameter save_best_only is set to true. This saves the model with the lowest validation loss. That model can than be used to make predictions.